A decomposition technique in integer linear programming

  • S. Giulianelli
  • M. Lucertini
Computational Techniques
Part of the Lecture Notes in Computer Science book series (LNCS, volume 41)


In this work, using the group theoretical approach we point out so me conditions on the B−1N matrix, often verified in practice, that make it possible to transform the system of linear congruences (constraints of problem 2) in a block diagonal form. In some cases, using this proce dure, the number of constraints can increase with respect to the number of constraints of problem 2. However, the problem can be solved indipen dently for the variables associated with each block.

This procedure leads to the indipendent solution of a number of subproblems in a smaller number of variables.

In the worst case each subproblem requires the same number of constraints as the original problem, but generally this number is smaller.


Integer Programming Integer Linear Programming Linear Programming Problem Mixed Integer Linear Programming Decomposition Technique 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1 J.J.H. FORREST, J.P.H. HIRST, J.A. TOULIN, Practical solution of large mixed integer programming problems with UMPIRE. Management Science, vol. 20, n. 5, January 1974.Google Scholar
  2. 2 G. MITRA, Investigation of some branch and bound algorithms for (0–1) mixed integer linear programming. Mathematical Programming 4, pp. 155–170, 1973.Google Scholar
  3. 3 M. SHAW, Review of computational experience in solving large mixed integer programming problems. pp. 406–412. Applications of Mathematical Programming Techniques, English Universities Press, London 1970.Google Scholar
  4. 4 A.M. GEOFFRION, G.W. GRAVES, Multicommodity distribution system design by Benders decomposition. Management Science, vol. 20, n. 5, January 1974.Google Scholar
  5. 5 G. GALLO, E. MARTINO, B. SIMEONE, Group optimization algorithms and some numerical results via a branch and bound approach. Giornate AICA su "Tecniche di Simulazione ed Algoritmi". Mila no, Nov. 1972.Google Scholar
  6. 6 J.F. SHAPIRO, Dynamic programming algorithms for the integer programming problem I: the integer programming problem viewed as a knapsack-type problem. Operation Research, 16 January 1968.Google Scholar
  7. 7 J.F. SHAPIRO, Group theoretic algorithms for the integer programming problem II: extension to a general algorithm. Operation Research 16, September 1968.Google Scholar
  8. 8 J.A. TOMLIN, Branch and bound methods for integer and non-convex programming. Integer and non-linear programming, cap. 21, North-Holland, Amsterdam 1970.Google Scholar
  9. 9 L.A. WOLSEY, Extensions of the group theoretic approach in integer programming. Management Science, vol. 18, n. 1, September 1971.Google Scholar
  10. 10 S. ZIONTS, Linear and integer programming, Prentice-Hall, 1974.Google Scholar
  11. 11 T.C. HU, Integer programming and network follows. Addison-Wesley Publishing Company, 1969.Google Scholar
  12. 12 R.E. GOMORY, Some polyhedra related to combinatorial problems. Linear Algebra and Its Applications, n. 2, 1969.Google Scholar
  13. 13 D.E. BELL, Improved bounds for integer programs: a supergroup approach. Research Memorandum of IIASA, November 1973.Google Scholar
  14. 14 H. GREENBERG, Integer programming. Academic Press, 1971.Google Scholar
  15. 15 A.M. GEOFFRION, Lagrangean relaxtion and its uses in integer programming. Western Management Science Institute, Working Paper n. 195.Google Scholar
  16. 16 M.L. FISHER, J.F. SHAPIRO, Constructive duality in integer program ming. Massachusetts Institute of Technology. Working Paper OK 008-72, April 1972.Google Scholar
  17. 17 R.S. GARFINKEL, G.L. NEMHAUSER, Integer programming. John Wiley and Sons, 1972.Google Scholar
  18. 18 G.S. MOSTOW J.H SAMSON, I.P. MEYER, Fundamental structure of algebra. Mc Graw-Hill, New York, 1963.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1976

Authors and Affiliations

  • S. Giulianelli
    • 1
  • M. Lucertini
    • 2
  1. 1.CSSCCA — CNRItaly
  2. 2.Istituto di AutomaticaUniversità di RomaRoma

Personalised recommendations